GATE 2025 EC Q.49: solution and answer

Official answer: CSource: Official answer key, GATE 2025 EC (IIT Roorkee)

1. Total charge on a surface. Integrate the surface charge density over the area: $Q={\iint }_S{\rho }_{s}dS$. The sheet lies in the $x$-$y$ plane, so $dS=dxdy$.

2. Limits from the figure. The corners are at $(\pm 2,\pm 2)$, so the sheet covers $-2\le x\le 2$ and $-2\le y\le 2$ (in m). That is a $4\text{m}\times 4\text{m}$ square, as the question says.

3. Separate the integral. ${\rho }_s=4|y|$ does not depend on $x$, so the double integral splits into a product: $$Q={\int }_{-2}^2{\int }_{-2}^{2}4|y|dxdy=4\left({\int }_{-2}^{2}dx\right)\left({\int }_{-2}^2|y|dy\right)$$

4. Integrate $|y|$ using symmetry. $|y|$ is an even function, so ${\int }_{-2}^2|y|dy=2{\int }_0^{2}ydy=2\cdot \frac{2^2}{2}=4{\text{m}}^2$. The $x$-integral is just the width of the sheet: ${\int }_{-2}^{2}dx=4\text{m}$.

5. Combine. With ${\rho }_s$ in $\mu {\text{C/m}}^2$: $$Q=4\times 4\times 4=64\mu \text{C}$$ This is already an integer, and it is option (C).